The Church has declared that these three Saints actually existed, However, because of the paucity of trustworthy historical material concerning their lives, she has seen fit to remove them from the liturgical calendar for the Universal Church and relegated them to local calendar.
In actuality, these three Saints were Roman Martyrs of the 2nd and 3rd century, buried in the cemetery of Praetextatus. Their names were woven into the legend of St. Cecilia, Virgin and Martyr, which became very popular from the 6th century on. But there is no historical reason to believe that they had anything to do with her.
PRAYER: Lord, we devoutly recall the sufferings of Sts. Tiburtius, Valerian, and Maximus. Give success to our joyful prayers and grant us also constancy in our Faith. Amen.